article
Fokker-Planck equations as scaling limits of reversible quantum systems
published
François
Castella
author
László
Erdös
author 4DBD5372-F248-11E8-B48F-1D18A9856A870000-0001-5366-9603
Florian
Frommlet
author
Peter
Markowich
author
We consider a quantum particle moving in a harmonic exterior potential and linearly coupled to a heat bath of quantum oscillators. Caldeira and Leggett derived the Fokker Planck equation with friction for the Wigner distribution of the particle in the large-temperature limit: however, their (nonrigorous) derivation was not free of criticism, especially since the limiting equation is not of Lindblad form. In this paper we recover the correct form of their result in a rigorous way. We also point out that the source of the diffusion is physically restrictive under this scaling. We investigate the model at a fixed temperature and in the large-time limit, where the origin of the diffusion is a cumulative effect of many resonant collisions. We obtain a heat equation with a friction term for the radial process in phase space and we prove the Einstein relation in this case.
Springer2000
Journal of Statistical Physics10.1023/A:1018667323830
1003-4543 - 601
yes
Castella, François, et al. “Fokker-Planck Equations as Scaling Limits of Reversible Quantum Systems.” <i>Journal of Statistical Physics</i>, vol. 100, no. 3–4, Springer, 2000, pp. 543–601, doi:<a href="https://doi.org/10.1023/A:1018667323830">10.1023/A:1018667323830</a>.
F. Castella, L. Erdös, F. Frommlet, P. Markowich, Journal of Statistical Physics 100 (2000) 543–601.
Castella, François, László Erdös, Florian Frommlet, and Peter Markowich. “Fokker-Planck Equations as Scaling Limits of Reversible Quantum Systems.” <i>Journal of Statistical Physics</i>. Springer, 2000. <a href="https://doi.org/10.1023/A:1018667323830">https://doi.org/10.1023/A:1018667323830</a>.
Castella F, Erdös L, Frommlet F, Markowich P. 2000. Fokker-Planck equations as scaling limits of reversible quantum systems. Journal of Statistical Physics. 100(3–4), 543–601.
Castella, F., Erdös, L., Frommlet, F., & Markowich, P. (2000). Fokker-Planck equations as scaling limits of reversible quantum systems. <i>Journal of Statistical Physics</i>. Springer. <a href="https://doi.org/10.1023/A:1018667323830">https://doi.org/10.1023/A:1018667323830</a>
F. Castella, L. Erdös, F. Frommlet, and P. Markowich, “Fokker-Planck equations as scaling limits of reversible quantum systems,” <i>Journal of Statistical Physics</i>, vol. 100, no. 3–4. Springer, pp. 543–601, 2000.
Castella F, Erdös L, Frommlet F, Markowich P. Fokker-Planck equations as scaling limits of reversible quantum systems. <i>Journal of Statistical Physics</i>. 2000;100(3-4):543-601. doi:<a href="https://doi.org/10.1023/A:1018667323830">10.1023/A:1018667323830</a>
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